Pauls decatonic

Hello Paul,

Yep I'm still hanging around, although I have been on holiday for 2 weeks so
I've had a lot of digests to trawl through !

I'll have a play with your scales and tell you what I think.

<Hello to all and this post is worth a month of not posting, so I hope
you will look at it:
...........................................................

I've been working hard on a new paper, and I've come up with some JI
intepretations of my two decatonic scales. For those unfamiliar with
them, they are meant to be the best 7-limit analogues of what the
diatonic scale is in the 5-limit. I'm only going to post 1 mode each
of three versions of each of the two scales -- you can work out the
other 9 rotations of each on your own, if you wish.

The canonical JI interpretation of the pentachordal decatonic scale
is:

Pitches Steps
1/1 21:20
21/20 160:147
8/7 21:20
6/5 10:9
4/3 21:20
7/5 15:14
3/2 16:15
8/5 15:14
12/7 21:20
9/5 10:9
2/1

Lattice:
4/3-------1/1-------3/2
\`. ,'/|\`. ,'/|\
\ 8/7-/-|-\12/7 / | \
\ | / 7/5------21/20\
\|/,' `.\|/,' `.\
8/5-------6/5-------9/5

Making use of the 50:49 comma, we can get these superparticular-step
versions . . .

Pitches Steps
1/1 15:14
15/14 16:15
8/7 21:20
6/5 10:9
4/3 21:20
7/5 15:14
3/2 16:15
8/5 15:14
12/7 21:20
9/5 10:9
2/1

Pitches Steps
1/1 21:20
21/20 16:15
28/25 15:14
6/5 10:9
4/3 21:20
7/5 15:14
3/2 16:15
8/5 15:14
12/7 21:20
9/5 10:9
2/1

The canonical JI interpretation of the symmetrical decatonic scale is:

Pitches Steps
1/1 15:14
15/14 49:45
7/6 15:14
5/4 16:15
4/3 15:14
10/7 21:20
3/2 10:9
5/3 21:20
7/4 15:14
15/8 16:15
2/1

Lattice:
5/3-------5/4------15/8
/|\`. ,'/|\`. ,'/
/ | \10/7-/-|-\15/14/
/ 7/6-------7/4 \ | /
/,' `.\|/,' `.\|/
4/3-------1/1-------3/2

Making use of the 50:49 comma, we can get these superparticular-step
versions, for those who care about such things (I'm thinking Justin
White, if you're still around):

Pitches Steps
1/1 21:20
21/20 10:9
7/6 15:14
5/4 16:15
4/3 15:14
10/7 21:20
3/2 10:9
5/3 21:20
7/4 15:14
15/8 16:15
2/1

and

Pitches Steps

1/1 15:14
15/14 10:9
25/21 21:20
5/4 16:15
4/3 15:14
10/7 21:20
3/2 10:9
5/3 21:20
7/4 15:14
15/8 16:15
2/1

There it is . . . these are periodicity blocks . . . Kraig, they're
CS, of course . . . Dan, are the superparticular-step ones
quadrivalent? Pierre, you must have special insight into this . . .
comments? Any other JI people out there . . . I want your points of
view . . . free to post your analyses. And I hope you strict-JI
people will try these . . . above all, I want to hear them used for
music . . .

One more thing . . . As most of you know, in the diatonic scale, you
need to use the 81:80 unison vector (aka syntonic comma) to avail
yourself of the scale's full triadic potential (six consonant 5-limit
triads). Well, for the pentachordal decatonic scale the 50:49 unison
vector is what is needed to avail yourself of the full tetradic
potential (six 'consonant' 7-limit tetrads) . . You need to use both
the 50:49 and 64:63 unison vectors (or one of these and 225:224) to
avail yourself of the full tetradic potential of the symmetrical
decatonic scale (eight 'consonant' 7-limit tetrads). . . Here are
fully "supplemented" versions (with enough comma variants to get all
the 'consonant' tetrads):

Pentachordal Decatonic:

1/1
21/20
8/7
6/5
4/3
7/5, 10/7
3/2
8/5
12/7, 42/25
9/5
2/1

Lattice:

7/5
/|\
/ | \
4/3-----/-1/1-\-----3/2
\`. /,'/|\`.\ ,'/|\
\ 8/7-/-|-\12/7 / | \
\ | / 7/5------21/20\
\|/,' \`.\|/,'/ `.\
8/5-----\-6/5-/-----9/5
\ | /
\|/
42/25

Symmetrical Decatonic:

1/1
21/20, 15/14
7/6, 25/21
5/4
21/16, 4/3
7/5, 10/7
3/2
5/3
7/4, 25/14
15/8, 40/21
2/1

25/21-----25/14
/|\ /|\
/ | \ / | \
/ 5/3-\---/-5/4-\----15/8
/,'/|\`.\ /,'/|\`.\ ,'/|
40/21/-|-\10/7-/-|-\15/14/ |
| / 7/6-------7/4------21/16
|/,' \`.\|/,'/ \`.\|/,'/
4/3-----\-1/1-/---\-3/2 /
\ | / \ | /
\|/ \|/
7/5------21/20

Music, music, music!!!>

Justin White

DAVID JONES LIMITED ACN 000 074 573

Thanks Justin. By the way, what is justintonation@hotmail.com?

"Paul H. Erlich" wrote:
>
> Thanks Justin. By the way, what is justintonation@hotmail.com?

It's an email address Paul.

People like you keep me in business. Wake up.

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm
* http://mp3.com/davidbeardsley

Paul wrote

<Thanks Justin. By the way, what is justintonation@hotmail.com?>

This is just my hotmail email address just a play on my first name and JI.

Although I have had alot of emails from people thinking that I am an 'authority'
on just intonation !

So it can be a little confusing.

Justin White

DAVID JONES LIMITED ACN 000 074 573